# What is (-\frac{1}{2}+\frac{i\sqrt{3}}{2})^{10}?

Oct 29, 2014

By rewriting in trigonometric form,

${\left(- \frac{1}{2} + \frac{\sqrt{3}}{2} i\right)}^{10} = {\left[\cos \left(\frac{2 \pi}{3}\right) + i \sin \left(\frac{2 \pi}{3}\right)\right]}^{10}$

by De Moivre's Theorem,

$= \cos \left(10 \cdot \frac{2 \pi}{3}\right) + i \sin \left(10 \cdot \frac{2 \pi}{3}\right)$

$= \cos \left(\frac{20 \pi}{3}\right) + i \sin \left(\frac{20 \pi}{3}\right)$

$= \cos \left(\frac{2 \pi}{3}\right) + i \sin \left(\frac{2 \pi}{3}\right)$

$= - \frac{1}{2} + \frac{\sqrt{3}}{2} i$

I hope that this was helpful.